This book is exactly what the title describes, an overview of how music and mathematics relate. We have articles written by a number of writers with varying perspectives. I will comment on each of those articles. But I can summarize my predominant “take-away” from this book. Yes, numbers describe universal aspects of music. Through ratios of string bisection and the overtones series, ratios that also pop up in wind instrument partials, we can see a mathematical basis for how we perceive music, a basis that could well be universal to any creature with the consciousness to perceive music.

But I also must point out that numbers themselves are symbols depicting an arithmetic or mathematical reality. When the symbols take on more importance than the reality they are utilized to depict, the symbols lose their effectiveness for describing reality. And when used in abstract formulas to try to create meaningful music, they risk creating nothing meaningful musically, or something only meaningful as a curiosity, like throwing paint up against a wall to see what it will look like. It might look great—more likely it will just look like a mess. Maybe with some tweaks here and there, music of this sort could be more compelling than a mess. But when composers turn the numbers and formulas into the creative vehicle, they are throwing paint against the wall, and it will sound like paint up against a wall most of the time.

Introductory Section) Music and mathematics: an overview – Susan Wollenberg

In Western intellectual tradition, music has been studied in the past as a science. Only recently has it been classified as an art. And mathematics has always been a huge part of that study of music.

Part I: Music and mathematics through history

1. Tuning and temperament: closing the spiral – Neil Bibby

This article goes into the formulas involved with various scales and intervals. We are reminded that equal temperament tuning does not become an issue unless we try to change tonal centers, to change the starting pitch of a scale—unless we decide to “change key”/“transpose.” We take a close look at ratios and scale intervals, and the mathematics of equal temperament tuning. We find out equal temperament tuning has been proposed in other cultures, such as medieval China. Numbers, in the form of ratios, guide us toward determining consonance and dissonance in music.

2. Musical cosmology: Kepler and his readers – J. V. Field

Kepler explored the mathematics of music and planetary orbits, finding simple ratios for both. Are simple ratios the universe’s way of organizing itself in physics, implying a metaphysical component, and do the ratios of music help bridge the gap between physics and metaphysics?

There is also a mention of the potoo bird, which sings a “descending diatonic scale.” This is evidence of a nonhuman creature utilizing a scale that humans would recognize, a scale that can be derived from the ratios the Greeks originally described. This is evidence of a universal musical reality.

In this article he focuses on the insights offered by the “oscillograph.”

A “real” sound, as opposed to a “synthesized sound” has irregularities in the patterns it reveals on the oscilloscope.

4. Faggot’s fretful fiasco – Ian Stewart

Another look at Pythagorean string bisection ratios—how they generate the tones of conventional scales, and how on fretted instruments, they lead to the necessity of an equal temperament tuning system.

5. Helmholtz: combinational tones and consonance – David Fowler

This is an examination of Herman von Helmholtz’s (1821-1894) attempts to set specific, quantifiable rules for consonance.

Helmholtz looks at all the math again on bisected string ratios and overtones, and tries to come up with rules on what is consonant and what is dissonant. The problem, as I see it, is that there are learned cultural customs with respect to music that make one person’s dissonance another person’s consonance. Some broad universal principles may be discernible. The octave appears to be universally consonant. Microtones appear to be universally dissonant. In between can depend on cultural context. A single note held for a long time can seem dissonant, if dissonance is a musical unpleasantness that begs for resolution. Or, a tritone, known in the West for some time as the “devil’s interval,” can be consonant as part of a blues chord—or a minor second or major seventh can be sweet sounding as part of a major seventh chord. But the Helmholtz ideas give us a basis for a discussion of consonance and dissonance, and can be very helpful if we resist the temptation to turn them into absolutes.

Part III: Mathematical structure in music

6. The geometry of music – Wilfrid Hodges

Discussions of musical lines, music on a page, and putting these lines through a geometric analysis. (I did not find this particularly helpful to the issues I am looking at.)

This chapter goes into detail on some 20th Century composers—Arnold Schoenberg, Anton Webern, Alban Berg, Pierre Boulez, Peter Maxwell Davies, Iannis Xenakis—and how these composers fashioned their compositions according to mathematical rules. I am familiar with many of these composers, and this chapter prompted my comment at the outset, about mathematical symbols used for the purpose of insight, and not as a primary method of composition. Frankly, and this is a matter of personal taste, I find much of the music derived using mathematical rules to be unsatisfying. Music creators still need to make choices for music to have a meaningful effect on an audience. As soon as they make their choices, they are no longer composing with numbers.

Part IV. The composer speaks

9. Microtones and projected planes – Carlton Gamer and Robin Wilson

Using geometrical planes to write music.

10. Composing with fractals – Robert Sherlaw Johnson

This is a discussion of using computers to write music. The writer asks if the computer wrote the music. Of course not! Whoever provided the computer with the instructions—the program, in computer parlance—wrote the music! And the article discusses choices that often need to be made to complete the musical piece. Of course, that is composer involvement! The choices, by the way, then take us back to the human being creating the sound. It is the sound that matters, the sound and what it brings to the listener. The program serves that purpose, and likely has a limited function, though in the hands of a skilled musical communicator, I can see computer music resulting in effective music. In fact, in creating “Seventh Hell,” from my CD “Issa Music,” I needed computer power to realize my sonic image of the 7/8 sections. A technically gifted performer probably could have played the synthesizer lines I use in that piece. But programming those lines on the computer, I think, made the piece much more effective. And computer power opens up new possibilities that would have been limited by technical capability in the past. But the composer, the music creator, still has to tell the computer what to do!